“I believe in everything until it’s disproved. So I believe in fairies, the myths, dragons.” — John Lennon
Our tour of subgroups of non–positively curved groups has reached its end.
What did we miss? Probably lots — apologies to any aggrieved authors out there.
Here are a couple of corners of the landscape that we are conscious of having failed to explore.
- Simplicial non–positive curvature. This page by Tomasz Elsner gives an introduction and many references.
- The recent work of Daniel Wise and his collaborators on special cube complexes. This includes the papers —
- The structure of groups with a quasi–convex hierarchy, and
- Cubulating small-cancellation groups (with Frederic Haglund).
(Arguably we touched upon some of the beginnings of this current hot topic.)
Most likely, there are many more.
of a hyperbolic group 
, the extension of the inclusion 
to a continuous 
is called a Cannon-Thurston map. By the continuity of 
, if such a map exists it’s unique.
, where 
is a hyperbolic closed 3-manifold fibering over a circle, and 
, where 
, a closed 2-manifold, is the fiber surface, there is a Cannon-Thurston map 
is an infinite hyperbolic normal subgroup of a hyperbolic group 
. Then there exists a Cannon-Thurston map 
be a finite graph of groups where all vertex groups 
and edge groups 
are hyperbolic, and all injections 
(where 
is incident to 
) are quasi-isometric embeddings. Suppose that 
is also hyperbolic. Then for any vertex 
, there is a Cannon-Thurston map 
.
be a proper embedding of proper hyperbolic spaces. Suppose, given 
, that there is a non-negative function 
such that 
as 
and for all geodesic segments 
outside an 
-ball around 
joining endpoints of 
is outside the 
(see the illustration below). Then there’s a Cannon-Thurston map 
.
does not extend continuously, there exist sequences 
in 
such that 
and 
for some 
, but 
and 
for 
in 
.![{d_Y(y_0, [x_m,y_m]) \rightarrow \infty}](http://s0.wp.com/latex.php?latex=%7Bd_Y%28y_0%2C+%5Bx_m%2Cy_m%5D%29+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0 "{d_Y(y_0, [x_m,y_m]) \rightarrow \infty}")
as 
. On the other hand, for 
large enough, ![{d_X(i(y_0), [i(x_m),i(y_m)])}](http://s0.wp.com/latex.php?latex=%7Bd_X%28i%28y_0%29%2C+%5Bi%28x_m%29%2Ci%28y_m%29%5D%29%7D&bg=ffffff&fg=000000&s=0 "{d_X(i(y_0), [i(x_m),i(y_m)])}")
stays uniformly close to ![{d_X(i(y_0),[u,v]) > 0}](http://s0.wp.com/latex.php?latex=%7Bd_X%28i%28y_0%29%2C%5Bu%2Cv%5D%29+%3E+0%7D&bg=ffffff&fg=000000&s=0 "{d_X(i(y_0),[u,v]) > 0}")
(see the discussion of visibility in the 
of hyperbolic metric spaces is a metric space 
admitting a map 
onto a simplicial tree 
satisfying 
, 
is path connected, rectifiable, and 
-hyperbolic. We denote the induced path metric by 
.
are uniformly proper. That is, there exists a function 
such that for all 
, if 
, then 
.
, 
is path connected and rectifiable. The induced path metric is denoted 
.
![{f_e: X_e \times [0,1] \rightarrow X}](http://s0.wp.com/latex.php?latex=%7Bf_e%3A+X_e+%5Ctimes+%5B0%2C1%5D+%5Crightarrow+X%7D&bg=ffffff&fg=000000&s=0 "{f_e: X_e \times [0,1] \rightarrow X}")
such that 
is an isometry onto 
.
and 
are 
-quasi-isometric embeddings into 
and 
, where 
and 
.
such that for any 
, then 
, and 
as 
.
. In order to shortcut an 
-geodesic from 
to 
, we can't use a path that travels to more than 
other vertex spaces 
, since the distance between such subspaces is at least 1. The length of a path in 
which goes to infinity like 
as 
increases.]
is glued to the upper vertex subspace by an isometry and to the lower vertex subspace by a 
-quasi-isometry stretching the metric by a factor of 2.
be a tree of hyperbolic metric spaces. If 
in 
is constructed containing 
, then 
is outside an 
. Applying the key lemma above finishes the proof.
intersecting 
is distance 1 away from 
when 
with 
, consider its universal cover 
(with the corresponding vertex and edge group assignments, so that 
; see below for an example). 
![\times [0,1]](http://s0.wp.com/latex.php?latex=%5Ctimes+%5B0%2C1%5D&bg=ffffff&fg=000000&s=0 "\times [0,1]")
and the corresponding quasi-isometric embeddings, we get a tree of hyperbolic spaces satisfying the hypotheses of Mitra’s theorem.
of hyperbolic groups to the boundaries 
. It is not evident that such maps should exist when 
. References include the textbooks of 
should be a Cantor set, which can be viewed (under the “obvious” embedding of the Cantor set in the plane – see picture below) as the set of limit points of 
. Then 
is the set of equivalence classes of geodesic rays 
originating at 
(i.e., 
). Ray 
is equivalent to 
, written 
, if 
, or equivalently if the images of 
is the set of equivalence classes of quasigeodesic rays (originating at any point in the space). Here we must use only the finite Hausdorff distance as the equivalence relation. [Note: this definition has no analogue for CAT(0) spaces, where there is no quasigeodesic stability.]
of generalized geodesic rays representing the points 
, i.e. geodesic segments connecting the basepoint ![{[0,\infty]}](http://s0.wp.com/latex.php?latex=%7B%5B0%2C%5Cinfty%5D%7D&bg=ffffff&fg=000000&s=0 "{[0,\infty]}")
by the constant map at 
So 
A subsequence converges to a geodesic ray from 
. See the illustration below.)
as 
for 
if they are represented by generalized rays (geodesic rays allowed) 
from 
has itself a subsequence converging to 
, a fundamental system of open sets in 
about 
is the collection of 
, where a generalized ray 
if 
. (The choice of 
does not matter since asymptotic rays from a fixed basepoint in a hyperbolic space are uniformly close: within 
of each other.)
as 
varies, induced by the projection maps to these complete convex subsets. The topology is the 
continuous).
denote the set of continuous functions from 
with the 
, where 
if 
is a constant map, by associating to 
the equivalence class of the distance to 
. The point of 
associated to a geodesic ray 
is represented by the Busemann function 
defined as
![\displaystyle b_c(x)=\limsup_{t\rightarrow\infty}{\left[d(x,c(t))-t\right]}.](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+b_c%28x%29%3D%5Climsup_%7Bt%5Crightarrow%5Cinfty%7D%7B%5Cleft%5Bd%28x%2Cc%28t%29%29-t%5Cright%5D%7D.&bg=ffffff&fg=000000&s=0 "\displaystyle b_c(x)=\limsup_{t\rightarrow\infty}{\left[d(x,c(t))-t\right]}.")
with the classical 
instead of near 0.
are represented by sequences in 
:
![\displaystyle (x\cdot y)_z=\frac12\left[d(x,z)+d(y,z)-d(x,y)\right].](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28x%5Ccdot+y%29_z%3D%5Cfrac12%5Cleft%5Bd%28x%2Cz%29%2Bd%28y%2Cz%29-d%28x%2Cy%29%5Cright%5D.&bg=ffffff&fg=000000&s=0 "\displaystyle (x\cdot y)_z=\frac12\left[d(x,z)+d(y,z)-d(x,y)\right].")
![{[z,x]}](http://s0.wp.com/latex.php?latex=%7B%5Bz%2Cx%5D%7D&bg=ffffff&fg=000000&s=0 "{[z,x]}")
and ![{[z,y]}](http://s0.wp.com/latex.php?latex=%7B%5Bz%2Cy%5D%7D&bg=ffffff&fg=000000&s=0 "{[z,y]}")
, or equivalently the distance from 
to ![{[x,y]}](http://s0.wp.com/latex.php?latex=%7B%5Bx%2Cy%5D%7D&bg=ffffff&fg=000000&s=0 "{[x,y]}")
. In a ![{d(z,[x,y])}](http://s0.wp.com/latex.php?latex=%7Bd%28z%2C%5Bx%2Cy%5D%29%7D&bg=ffffff&fg=000000&s=0 "{d(z,[x,y])}")
(to within some fixed multiple of 
is said to converge at infinity if 
as 
, and two such sequences 
converge to the same point if 
. (Transitivity of this relation follows from one of the many equivalent definitions of 
for all 
.) A fundamental system of neighborhoods of a boundary point is given by bounding the Gromov overlap from below by a sequence of numbers approaching infinity. 
, 
, there is a geodesic 
with 
, 
. Indeed, take geodesic rays 
originating from a basepoint 
. Since 
(shown in blue) a distance more than 
. For 
, we can choose geodesic segments 
to 
, and extend them to generalized geodesic lines. By 
, which we can assume is 
(shown in red). Then Arzelà–Ascoli gives a subsequence of 
is within 
for 
and within 
of 
. (See the diagram below.) So 
between hyperbolic spaces, there is at most one way to extend it continuously to the boundary 
.
is a (continuous) embedding, where 
(using the quasigeodesic model 
of hyperbolic boundary.) If 
is a quasi-isometry, then 
, so 
unless 
.
to a hyperbolic group 
and 
.
is a Cantor set.![{M=(S\times[0,1])/((x,0)\sim(\phi(x),1))}](http://s0.wp.com/latex.php?latex=%7BM%3D%28S%5Ctimes%5B0%2C1%5D%29%2F%28%28x%2C0%29%5Csim%28%5Cphi%28x%29%2C1%29%29%7D&bg=ffffff&fg=000000&s=0 "{M=(S\times[0,1])/((x,0)\sim(\phi(x),1))}")
of a hyperbolic surface 
is a hyperbolic 3-manifold if and only if 
is an HNN-extension of a surface group 
, with
, a space-filling curve. Therefore 
via its inclusion in 
: the set of accumulation points in 
of any 
Since 
of 
. Therefore 
(The interior of a ball together with some points on its boundary is alway convex, after all.) So 
.
. A projective resolution of a 
is an exact sequence 
is a projective 
as a trivial 
of a group 
such that 
have the property that their finitely presented subgroups are also hyperbolic. This result appears in Gersten’s paper 
is a group and 
is a finitely generated 
–module. Let 
be a finitely generated free 
, such that there is a surjective homomorphism 
of 
is a free 
–basis for 
–norm
.
. Such a norm 
on 
and 
is obtained from another surjective homomorphism from a different finitely generated free 
–module 
, then there is a uniform constant 
such that 
for all 
. So 
is free and finitely generated and given an 
associated to some basis, and
is projective.
for 
(that is, a map 
is the identity on 
so that 
for all 
.
is projective, we know that 
. We choose a map 
such that 
is the identity on 
be a finitely generated free submodule generated by a subset of the generators of 
which contains the image of 
. Let 
. Notice that 
is also free, but not necessarily finitely generated. We can modify the retraction 
to 
and 
. Let 
be a surjective homomorphism of 
such that 
. This map 
can be extended to 
by setting the images under 
. It follows this 
and 
.
are both finitely generated and free. So the map 
for all 
. Since the standard norm on 
for all 
. This completes the proof.
be an arbitrary group and let 
. Then there exists an 
-dimensional 
-complex 
. If 
-complex.
be a finitely presented group. So 
, where 
and 
is the normal closure of the relations 
. Let the presentation 
–complex of this group be 
, the abelianization of R , and the 
–action on it is induced by the conjugation of 
–module structure on 
, where 
is the 
-skeleton of the universal cover of the presentation complex 
–complex 
.
is an exact sequence of 
projective, then 
is projective.
are projective and 
is projective.
–modules 
. By definition this means that 
such that 
as 
free as a 
of 
and 
satisfy 
. Further, by adding trivial letters and corresponding relations to the presentation for 
. An application of the 
is projective as a 
. 
for 
, such that there is a uniform constant 
with the property that 
for all 
. So if 
, then 
. 
be a circuit in 
and let 
be a 
of minimal area. The area of 
, where 
is the element of 
corresponding to 
, where now 
is in terms of 
as a 
satisfies a linear isoperimetric inequality, then so does 
. Since 
. Hyperbolic groups are always of type 
is a branching locus if it satisfies the following two conditions:
-cell 
of 
, we have that 
is a disjoint union of faces of 
.
is nonempty and connected for each cell 
.
, the link 
is a full subcomplex of 
.
is obtained as follows:
.
. If 
and let 
.
, which is an example of a branching locus.
: 
is free of rank 
. As a free basis, take the six loops 
. (The bars indicate paths traversed in the opposite direction to their orientations.) Let 
be the homomorphism such that
and 
are the permutations 
and 
, respectively. The essential feature of this construction is that 
maps all commutators of 
or 
with 
or 
to non–trivial five–cycles, since 
. 
, let 
be the projections 
, 
and 
. All are continuous, onto, and induce homomorphisms 
. 
be a retract 
to 
. For 
. 
defines a 
-fold covering of 
be the natural map 
.
is hyperbolic
space. By the following two theorems, to establish that 
–hyperbolic metric space. Then 
is a complete bipartite graph where each component has four vertices (along with some half edges i.e. edges with one end–point missing). Each loop is this graph that consists of four edges maps to the boundary of a neighborhood of a point in 
under exactly one of the maps 
or 
. Such a loop deformation retracts to one of the commutator loops in 
. Therefore in the branched cover, the link structure around these vertices is distorted, i.e. the aforementioned loops lift to 
–fold copies.
in the universal cover intersecting this plane that is not parallel to it, which maps to an edge in the branching locus. Let the intersection of the flat plane with this edge be 
. The link of 
give rise to a map to the oriented unit circle mapping the edges to the cirlce respecting the orientation. This extends linearly to a map 
.
lifts to a map from the universal cover of 
. The ascending and descending links of any vertex in this Morse theory setting are 
induced by 
, and in particular not hyperbolic. For further details of such Combinatorial Morse theory, see 
but not of type 
be a presentation complex for some presentation of 
is a loop in 
, we define the filling area of 
, or equivalently if and only if 
is subquadratic. This result is due to Gromov; the absence of Dehn functions between linear and quadratic is called the “Gromov Gap”.
. (On account of the Gromov gap, this means that 
or 
.)
.
such that 
is a short exact sequence, 
.
and is defined by 
. The automorphism 
is given by 
and 
. Let 
, the free group generated by 
and 
. You can see that 
by considering the family of words 
(with length on the order of 
letters, and it is clearly reduced in 
, where 
. The corresponding presentation complex built out of squares can be shown to satisfy the link condition.
of 
. 
. Alternatively, see 
, for a proof see Theorem 6.20 in 
. Now consider the following family of embedded disks in 
:
. Moreover, the boundary loops of these disks do not admit fillings with smaller area, since 
-dimensional and aspherical. (In such spaces the embedded filling is the most efficient among the fillings of any loop.) This shows that 
is a lower bound on the Dehn function. 
embeds in 
. Since 
. Then the subgroup 
of 
. (It is easy to see that the relations 
and 
, and the corresponding relations involving 
and 
in Step 1, one can construct other examples.
, where 
is a CAT(0) group, and 
. (The example above is the case n=2.) Then the double 
embeds in 
, and 
. Thus, 
occurs as the Dehn function of a subgroup of a CAT(0) group for all integers 
. 
be a hyperbolic surface bundle over a circle. Then one has a short exact sequence 
, and 
is a CAT(0) group. Bridson and Haefliger prove that if 
. Hence, 
and 
. In fact it is 
. Thus, 
occurs as the Dehn function of a subgroup of a CAT(0) groups.
to use in Step 1, for example due to 
for 
or functions growing faster than 
.
. The lower bound was proved in a manner similar to the example above, by constructing a sequence of embedded diagrams in the level set for 
.
, where 
is not an integer, but 
has Dehn function 
. 
. Using these in Step 1 of the Bieri doubling trick, they produce a class of examples of subgroups of CAT(0) groups with Dehn functions of the form 
is an integer. The kernels are of type 
, unlike the original Bieri doubles, which all have finite 
‘s. Another interesting direction is to study higher dimensional Dehn functions of CAT(0) groups and their subgroups. We won’t go into the technicalities here, but just say that the 
measures the difficulty of filling 
-chains (or 
. (As in the case 
, the group 
attains this general upper bound.) 
, such that the membership problem for 
such that there is no algorithm to decide membership of 
has a solvable conjugacy problem (
where
is finitely presented and 
, and so questions about equality in 
), 
, can be identified with to a finite presentation 
of 
is finitely generated as a 
is finite. This can be seen by taking the attaching maps of the 3-cells as the generators of the module. This identification allows one to find a nice presentation for 
be a presentation for a group 
be a sequence of words of the form 
where 
and 
is some word over 
. We call such a sequence an identity sequence if 
is freely equal to the empty word in 
. An equivalence relation is given on identity sequences, and we consider the action of 
. This action naturally induces a 
. Thus 
, a compact, negatively curved, piecewise hyperbolic 2-dimensional complex 
,
,
of cardinality at least 2,
is represented by one of the 1-cells of 
follows from the next result.
such that 
and unsolvable word problem. The enhanced Rips construction is applied to to 
, where 
where 
and the 
are lifts of generators of 
for a generating set for 
is in 
be finitely generated groups. Suppose 
and there exists 
such that 
. If there is no algorithm to decide membership of 
where 
, as item (9) of Theorem 3 tells us that the centralizer of 
is 
, and thereby gives us unsolvability of the conjugacy problem.
and 
is the fundamental group of a compact non=-positively curved squared complex, and is thus biautomatic (
such that there is no algorithm to determine whether or not 
is isomorphic to 
. 
where 
is an HNN-extension of 
where 
generates a maximal cyclic subgroup.
